„Judged by the only standards which are admissible in a pure doctrine of numbers i is imaginary in the same sense as the negative, the fraction, and the irrational, but in no other sense; all are alike mere symbols devised for the sake of representing the results of operations even when these results are not numbers (positive integers).“

Fonte: The Number-System of Algebra, (1890), p. 86; Reported in Moritz (1914, 282)

Última atualização 4 de Junho de 2020. História

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„It may be in some measure due to the defects of notation in his time that Diophantos will have in his solutions no numbers whatever except rational numbers, in [the non-numbers of] which, in addition to surds and imaginary quantities, he includes negative quantities. …Such equations then as lead to surd, imaginary, or negative roots he regards as useless for his purpose: the solution is in these cases ὰδοπος, impossible. So we find him describing the equation 4=4x+20 as ᾰτοπος because it would give x=-4. Diophantos makes it throughout his object to obtain solutions in rational numbers, and we find him frequently giving, as a preliminary, conditions which must be satisfied, which are the conditions of a result rational in Diophantos' sense. In the great majority of cases when Diophantos arrives in the course of a solution at an equation which would give an irrational result he retraces his steps and finds out how his equation has arisen, and how he may by altering the previous work substitute for it another which shall give a rational result. This gives rise, in general, to a subsidiary problem the solution of which ensures a rational result for the problem itself. Though, however, Diophantos has no notation for a surd, and does not admit surd results, it is scarcely true to say that he makes no use of quadratic equations which lead to such results. Thus, for example, in v. 33 he solves such an equation so far as to be able to see to what integers the solution would approximate most nearly.“

—  Thomas Little Heath British civil servant and academic 1861 - 1940

Diophantos of Alexandria: A Study in the History of Greek Algebra (1885)

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„The transfinite numbers are in a certain sense themselves new irrationalities“

—  Georg Cantor mathematician, inventor of set theory 1845 - 1918

As quoted in Understanding the Infinite (1994) by Shaughan Lavine
Contexto: The transfinite numbers are in a certain sense themselves new irrationalities and in fact in my opinion the best method of defining the finite irrational numbers is wholly dissimilar to, and I might even say in principle the same as, my method described above of introducing transfinite numbers. One can say unconditionally: the transfinite numbers stand or fall with the finite irrational numbers; they are like each other in their innermost being; for the former like the latter are definite delimited forms or modifications of the actual infinite.

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„Since it is only in the context of a proposition that words have any meaning, our problem becomes this: To define the sense of a proposition in which a number-word occurs.“

—  Gottlob Frege mathematician, logician, philosopher 1848 - 1925

Nur im Zusammenhange eines Satzes bedeuten die Wörter etwas. Es wird also darauf ankommen, den Sinn eines Satzes zu erklären, in dem ein Zahlwort vorkommt.
Gottlob Frege (1950 [1884]). p. 73

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„Marcel Duchamp's 'Nude descending the stairs' is the result of the desire for motion. Here he has also eliminated representative form. This avoids the connotation of ideas which would interfere with the success of the main issue - the sense of movement.“

—  Alexander Calder American artist 1898 - 1976

1930s - 1950s, Statement from Modern Painting and Sculpture', (1933)
Fonte: en.wikiquote.org - Alexander Calder / Quotes of Alexander Calder / 1930s - 1950s / Statement from Modern Painting and Sculpture', (1933)

E. W. Hobson photo

„Perhaps the least inadequate description of the general scope of modern Pure Mathematics—I will not call it a definition—would be to say that it deals with form, in a very general sense of the term; this would include algebraic form, functional relationship, the relations of order in any ordered set of entities such as numbers, and the analysis of the peculiarities of form of groups of operations.“

—  E. W. Hobson British mathematician 1856 - 1933

Fonte: Presidential Address British Association for the Advancement of Science, Section A (1910), p. 287; Cited in: Robert Edouard Moritz. Memorabilia mathematica; or, The philomath's quotation-book https://archive.org/stream/memorabiliamathe00moriiala#page/4/mode/2up, (1914), p. 5: Definitions and objects of mathematics.

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„The hospitality of primitive peoples, respect for human life, the sense of reciprocal obligation, compassion for the weak, courage, extending even to the sacrifice of self for others which is first learnt for the sake of children and friends, and later for that of members of the same community — all these qualities are developed in man anterior to all law, independently of all religion, as in the case of the social animals.“

—  Peter Kropotkin Russian zoologist, evolutionary theorist, philosopher, scientist, revolutionary, economist, activist, geographer, writer 1842 - 1921

Fonte: Law and Authority (1886), II
Contexto: The hospitality of primitive peoples, respect for human life, the sense of reciprocal obligation, compassion for the weak, courage, extending even to the sacrifice of self for others which is first learnt for the sake of children and friends, and later for that of members of the same community — all these qualities are developed in man anterior to all law, independently of all religion, as in the case of the social animals. Such feelings and practices are the inevitable results of social life. Without being, as say priests and metaphysicans, inherent in man, such qualities are the consequence of life in common.
But side by side with these customs, necessary to the life of societies and the preservation of the race, other desires, other passions, and therefore other habits and customs, are evolved in human association. The desire to dominate others and impose one's own will upon them; the desire to seize upon the products of the labor of a neighboring tribe; the desire to surround oneself with comforts without producing anything, while slaves provide their master with the means of procuring every sort of pleasure and luxury — these selfish, personal desires give rise to another current of habits and customs.